Energy spectrum and the quantum Hall effect on the square lattice with next-nearest-neighbor hopping

Abstract

We study the energy spectrum and the Hall effect of electrons on the square lattice with next-nearest-neighbor (NNN) hopping as well as nearest-neighbor hopping. This lattice includes the triangular lattice as a special case. We study the system under general rational values of magnetic flux per unit cell \ensuremath{\varphi}=p/q. The structure of the secular equation is studied in detail, and the k dependence of the energy is analytically obtained. In the absence of NNN hopping, the two bands at the center touch for q even; thus the Hall conductance is not well defined at half-filling. An energy gap opens there by introducing NNN hopping. When \ensuremath{\varphi}=1/2, the NNN model coincides with the mean-field Hamiltonian for the chiral spin state proposed by Wen, Wilczek, and Zee [Phys. Rev. B 39, 11 413 (1989)]. At half-filling for q even, the Hall conductance is calculated from the Diophantine equation and the E-\ensuremath{\varphi} diagram. We also explicitly calculate the Hall conductance for \ensuremath{\varphi}=1/2 using the wave function. We find that gaps close for other fillings at certain values of NNN hopping strength. The quantized value of the Hall conductance changes once this phenomenon occurs.